The moment of inertia experiment for a rigid body is a physics experiment in which the angular acceleration produced by an external force is measured to determine the moment of inertia, a quantity that represents resistance to rotational acceleration.
In linear motion, an object’s mass represents how difficult it is to change its state of motion. In rotational motion, however, resistance to rotation depends not only on the total mass but also on how that mass is distributed relative to the axis of rotation. The quantity that describes this resistance to rotational motion is the moment of inertia.
In an experimental report, it is important not only to compare the measured moment of inertia with the theoretical value, but also to discuss how bearing friction, string slippage, the radius of the rotation shaft, falling distance, time measurement, and the moment of inertia of the apparatus itself affected the result.
This article assumes an experiment in which a falling mass rotates a rigid body and summarizes the items to record in the results, sample experimental values, calculation methods, sources of error, example results statements, and discussion examples that can be used in a report.
Note:
This article is a study reference for discussing results obtained in physics experiments at universities and similar institutions. The experimental values shown here are illustrative examples and are not actual measured results.
For assembly of the rotating apparatus, securing the falling mass, winding the string, measurement ranges, and safety precautions, always follow the laboratory manual and the instructions of your instructor or teaching assistant.
What Is the Moment of Inertia of a Rigid Body?
The moment of inertia is a quantity that represents how difficult it is to rotate an object about a particular axis. It is generally denoted by I, and its unit is kg·m2.
For a point mass of mass m at a distance r from the axis of rotation, the moment of inertia is expressed as follows.
I = mr2
For a rigid body, the moment of inertia is determined by considering the mass of each part of the object and its distance from the axis of rotation. Therefore, even for objects with the same total mass, an object whose mass is distributed farther from the axis has a larger moment of inertia.
In rotational motion, torque τ, moment of inertia I, and angular acceleration α are related by:
τ = Iα
When a rotating body is accelerated by a falling mass, its moment of inertia can be determined by considering both the translational motion of the falling mass and the rotational motion of the rotating body.
Main Items to Record in the Results
The experimental results should include not only the final moment of inertia but also the apparatus conditions and raw data used in the calculations.
- Type and shape of the rotating body
- Mass of the rotating body
- Radius or length of the disk, ring, rod, or other object
- Radius of the rotation shaft or winding shaft
- Mass of the falling weight
- Falling distance
- Falling time
- Number of measurements for each condition
- Average falling time
- Acceleration of the falling mass
- Angular acceleration of the rotating body
- Moment of inertia obtained experimentally
- Moment of inertia calculated from the theoretical formula
- Difference and relative error between the experimental and theoretical values
- Moment of inertia of the apparatus alone
- Any string slippage, vibration, or irregular rotation observed during measurement
When several rigid bodies are measured, comparing not only their masses and shapes but also how their mass is distributed relative to the rotation axis makes it easier to discuss the physical meaning of moment of inertia.
Sample Experimental Conditions
| Item | Sample condition |
|---|---|
| Measurement method | Wind a string around the rotation shaft and accelerate the rotating body by allowing a mass attached to the end of the string to fall |
| Object measured | Uniform disk |
| Disk mass | 1.20 kg |
| Disk radius | 0.100 m |
| Winding shaft radius | 0.0100 m |
| Falling mass | 0.0500 kg |
| Falling distance | 0.800 m |
| Number of measurements | 3 |
| Gravitational acceleration | 9.80 m/s2 |
| Theoretical formula for a disk | I = 1/2 MR2 |
In an actual apparatus, the rotation shaft, supports, mounting hardware, and other components also have moments of inertia. Therefore, the moment of inertia of the apparatus alone may be measured and subtracted from the value obtained with the rigid body installed.
Sample Experimental Values
The following are sample values for determining the moment of inertia of a uniform disk using a falling mass.
| Trial | Falling distance h (m) | Falling time t (s) | Acceleration a (m/s2) | I of entire apparatus (kg·m2) |
|---|---|---|---|---|
| 1 | 0.800 | 14.62 | 0.00749 | 0.00654 |
| 2 | 0.800 | 14.68 | 0.00743 | 0.00659 |
| 3 | 0.800 | 14.65 | 0.00745 | 0.00657 |
Average Moment of Inertia of the Entire Apparatus
Average = (0.00654 + 0.00659 + 0.00657) / 3
= 0.00657 kg·m2
If the moment of inertia of the apparatus alone is 0.00058 kg·m2, the experimental value for the disk alone is:
Idisk = Itotal − Iapparatus
= 0.00657 − 0.00058
= 0.00599 kg·m2
The theoretical value for the disk is 0.00600 kg·m2, so the experimental value is close to the theoretical value.
How to Calculate the Moment of Inertia
Acceleration of the Falling Mass
Assuming the falling mass starts from rest and falls with constant acceleration, the falling distance h and falling time t are related by:
h = 1/2 at2
Therefore, the acceleration a is:
a = 2h / t2
For a falling distance of 0.800 m and a falling time of 14.65 s:
a = 2 × 0.800 / 14.652
= 1.600 / 214.62
= 0.00745 m/s2
Angular Acceleration
If the string winds around the shaft without slipping, the acceleration a of the falling mass, shaft radius r, and angular acceleration α are related by:
α = a / r
For a winding shaft radius of 0.0100 m:
α = 0.00745 / 0.0100
= 0.745 rad/s2
Moment of Inertia of the Entire Apparatus
Let m be the falling mass, g gravitational acceleration, a the falling acceleration, and r the winding shaft radius. If friction is neglected, the moment of inertia is:
I = mr2(g / a − 1)
Substituting the sample values:
I = 0.0500 × 0.01002 × (9.80 / 0.00745 − 1)
= 0.00000500 × 1314.4
= 0.00657 kg·m2
In actual university experiments, however, the calculation formula may differ depending on the structure of the shaft and pulley, the string winding position, and apparatus-specific correction formulas. The equation above is a simplified expression of the basic relationship.
The sample table illustrates a case in which the moment of inertia of the entire apparatus is calculated using an experimental formula that includes correction terms based on the apparatus specifications and the effect of friction. In an actual report, always use the equation specified in your laboratory manual.
Theoretical Value for the Disk
For a uniform disk of mass M and radius R rotating about an axis through its center and perpendicular to its plane, the theoretical moment of inertia is:
I = 1/2 MR2
For a mass of 1.20 kg and radius of 0.100 m:
I = 1/2 × 1.20 × 0.1002
= 0.00600 kg·m2
Relative Error
If the experimental value is 0.00599 kg·m2 and the theoretical value is 0.00600 kg·m2, the relative error is:
Relative error (%) = |experimental value − theoretical value| / theoretical value × 100
= |0.00599 − 0.00600| / 0.00600 × 100
= approximately 0.17%
Main Sources of Error
| Source of error | Effect on measurement | Likely effect on result | Improvement/check |
|---|---|---|---|
| Friction in the rotation shaft or bearings | Part of the force from the falling mass is used to overcome friction | Acceleration decreases, tending to overestimate the moment of inertia | Perform a free-rotation measurement and correct for frictional torque |
| String slippage | The relationship between falling acceleration and angular acceleration breaks down | Produces scatter or systematic deviation | Wind the string uniformly without overlapping |
| String stretching | Changes the relationship between falling distance and rotation angle | Causes error in converting acceleration to angular acceleration | Use a low-stretch string |
| Error in measuring shaft radius | Directly affects calculations of angular acceleration and torque | Overestimating the radius tends to overestimate the moment of inertia | Measure in several directions with calipers |
| Error in falling-time measurement | Affects calculation of falling acceleration | Measuring too long a time decreases acceleration and tends to increase I | Use a photoelectric sensor and repeat measurements |
| Error in falling-distance measurement | Affects calculation of falling acceleration | Overestimating the distance increases the calculated acceleration | Clearly define the start and end points |
| Eccentric rotation axis | Produces vibration and periodic resistance during rotation | Increases scatter in measured times | Mount the rotating body correctly at the center of the axis |
| Error in correcting for the apparatus moment of inertia | Prevents accurate isolation of the rigid body’s moment of inertia | Shifts all experimental values systematically | Measure the apparatus alone under the same conditions |
| Swinging of the falling mass | The force no longer acts purely vertically | Causes variation in falling time | Release the mass gently and keep the string vertical |
Why Neglecting Friction Causes Error
In an ideal calculation, the gravitational potential energy of the falling mass is assumed to be converted into translational kinetic energy of the falling mass and rotational kinetic energy of the rotating body.
In reality, some energy is converted into heat by bearing friction and air resistance. As a result, the falling acceleration is smaller than its ideal value. If friction is ignored, the rotating body may appear more difficult to rotate than it actually is, causing the moment of inertia to be overestimated.
Why the Rotation-Shaft Radius Is Important
The winding shaft radius affects both the torque produced by the falling mass and the angular acceleration of the rotating body. Because some formulas contain the square of the radius, even a small measurement error can have a relatively large effect on the calculated moment of inertia.
If the string overlaps in multiple layers on the shaft, the effective radius changes during the measurement. It is therefore important to wind the string uniformly in a single layer.
Why Mass Distribution Changes the Moment of Inertia
The moment of inertia depends not only on the mass of each part but also on how far that mass is from the rotation axis.
Mass farther from the rotation axis contributes more strongly to the moment of inertia in proportion to the square of its distance. Therefore, even with the same total mass and outer radius, a ring with mass concentrated near the outside has a larger moment of inertia than a disk whose mass is distributed uniformly throughout its interior.
Example of How to Write the Results
A disk was rotated using a falling mass, and the moment of inertia of the entire rotating apparatus was determined from the falling distance and falling time.
Three measurements were performed with a falling distance of 0.800 m. The moment of inertia of the entire apparatus ranged from 0.00654 to 0.00659 kg·m2, with an average of 0.00657 kg·m2.
After subtracting the moment of inertia of the apparatus alone, 0.00058 kg·m2, the moment of inertia of the disk was 0.00599 kg·m2. The theoretical value calculated from the disk’s mass and radius was 0.00600 kg·m2, giving a relative error of approximately 0.17%.
Points to Connect to the Discussion
- How close was the experimental value to the theoretical value?
- What was the relative error?
- Was the moment of inertia of the apparatus itself subtracted appropriately?
- In which direction would neglecting bearing friction affect the result?
- Did the string slip or overlap while winding?
- Was the winding shaft radius measured accurately?
- Did the falling mass move vertically?
- Was the rotating body correctly centered on the rotation axis?
- Were the falling distance and time measured with sufficient precision?
- Can the relationship between mass distribution and moment of inertia be explained?
- Can differences among the theoretical formulas for disks, rings, and rods be explained?
Discussion Example
In this experiment, a disk was rotated by a falling mass, and the moment of inertia of the rigid body was determined from the falling distance and falling time. After subtracting the moment of inertia of the apparatus itself, the experimental value for the disk was 0.00599 kg·m2. The relative error with respect to the theoretical value of 0.00600 kg·m2 was approximately 0.17%. Thus, the experimental value agreed well with the theoretical value.
The moment of inertia of a disk depends not only on its mass and radius but also on how the mass is distributed relative to the rotation axis. For the disk used here, the mass is uniformly distributed from the center to the outer edge, so the moment of inertia about the central axis is expressed as 1/2 MR2. The close agreement between the experimental value and this theoretical expression confirms the effect of mass distribution in rotational motion.
Possible reasons for the small discrepancy between the experimental and theoretical values include bearing friction, string slippage, error in measuring the winding shaft radius, error in measuring the falling time, and error in correcting for the moment of inertia of the apparatus itself.
In particular, when friction is present in the bearings, part of the torque produced by the falling mass is used to overcome that friction. Consequently, the falling mass accelerates less than it would ideally. If friction is ignored in the calculation, the rotating body appears more difficult to rotate than it actually is, which can cause the moment of inertia to be overestimated.
The winding shaft radius also affects calculations of both torque and angular acceleration. If the string overlaps on the shaft, the effective radius changes during the measurement, violating the assumption of a constant radius. Furthermore, if the string slips, the relationship between the falling acceleration and the angular acceleration of the rotating body no longer holds, producing a systematic error in the calculated moment of inertia.
To improve measurement accuracy, the frictional torque of the rotation axis should be measured separately and corrected for, and the string should be wound uniformly in a single layer. It is also effective to measure the shaft radius several times with calipers and to measure the falling time automatically using a photoelectric sensor or similar device.
In addition, performing similar measurements for a ring and a rod as well as a disk would make it possible to compare how the moment of inertia increases when mass is distributed farther from the rotation axis, even for objects with the same total mass. This follows from the fact that moment of inertia is determined by each mass element and the square of its distance from the rotation axis.
Discussion Example When the Experimental Value Is Larger Than the Theoretical Value
If the experimental value is larger than the theoretical value, one possible cause is insufficient correction for friction in the rotation shaft or bearings. Friction reduces the acceleration of the falling mass, making the rotating body appear more difficult to rotate and causing the calculated moment of inertia to be larger. The experimental value may also become larger if the falling time is measured longer than its actual value or if the winding shaft radius is overestimated.
Discussion Example When the Experimental Value Is Smaller Than the Theoretical Value
If the experimental value is smaller than the theoretical value, possible causes include subtracting too large a value for the moment of inertia of the apparatus itself or measuring the falling time shorter than its actual value. If the string slips and the motion of the falling mass is not fully transmitted to the rotating body, the assumptions underlying the calculation formula also break down, which can produce a smaller experimental value.
Summary
In a rigid-body moment-of-inertia experiment, a falling mass applies torque to a rotating body, and the falling distance and time are used to determine acceleration, angular acceleration, and ultimately the moment of inertia.
In the discussion, it is important to explain not only the difference between the experimental and theoretical values but also the effects of mass distribution, distance from the rotation axis, bearing friction, string slippage, winding shaft radius, time measurement, and correction for the apparatus itself.
By comparing rigid bodies of different shapes, such as disks, rings, and rods, it is possible to confirm that the moment of inertia increases as more mass is distributed farther from the rotation axis.
